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ATOMICTHEORY FOR STUDENTS OF METALLURGY WILLIAM HUME-ROTHERY， M.A.，D.Sc.，F.R. S. Royal Society Warren Research Fellow, .Lecturer in' Metallurgical Chemistry, University o, f'·Oxford Institute of Metals Monograph and Report Series No. 3 11948
GREEK ALPHABETo Rho Sigma. Tau Upsilon Phi Chi Psi Omega P口了vl甲JX访才田 P万T丫中X甲0 Iota .Kappa Lambda Mu Nu Xi Omicron Pi K入拼F亡‘0叮 IKAMN︻昌On Alpha Beta Gamma Delta Epsilon Zeta Eta Theta, 住召J夕6 石分﹄ 刃0口 ABr△EZHO CORRIGENDUM It is regretted that a sentence on page 4_5 of -he “The Structure ofMetals and Alloys Theory for Students of R-Setallurgy" ”and on page 7 of author's "Atomic is ambiguous 丁he kx 、Vhen a unit of length is larger than the true An or incorrect. gstrom unit. length is expressed ;n LY‘、，;，。+-P,。。，，，.。，犷r-,7，，。74,11.2 ;。 11L A.CX.‘名AJLLO CL压气./L l41/&C，{V{}&、‘侣份L 41L,孟‘〕 thus smaller than when the length is expressed in true Angstrom units, order and all values in kX. units require multiplying by 1.00202 in to give the values in true Angstrom units. 1 Crystal Angstrorn二1 kX. unit二1.OJ2LY2 x 10 We have thus: 一gcm. VI
Ct}NTENTS. 于盛0名 朽i 11248穷51670er心212日口 11123盛盛心肠567889 I,--Ts$0MUM" CHAPTxx BAOXOsov*in -General Introduction .一The First Theory of the Hydrogen Atom .一习、.Theory of Bohr ，一The Unoertainty Principle of Heisenberg Vn Ideas of Wave Mechanics Equations and Their Interpretation， Lll.m.rV.v.VI. nta幼on of the Individual VIII. The Meaning of Velocity。 IX.--Potential Boundaries in Wave Mechanics If.---Tnx STRuaruRE or THE FRxs ATox CHAPTx R‘X.-The Electron Groups XL--Methods of Representation of the Elootron- Cloud Density XII.---The a, p, and d States IIL---As9EMswxs or ATows CHArrim XIII. The Soft X-Ray Spectra of Solids XIV.-The Elements二 XV.---Assemblics of Unlike Atoms XV I.-Atomic Attraction and the Nature of van der Waa}srOrem 146 121 XVIL The of the Co-Valent Bond and th of Exch ange Forces XVIII.--Directed Valencies XIX.---Resonance Bonding and L如kage. the Metallic 1邓 136 PART IV. THz F$E1C-ELEC'1'RON THEony or MWAL3 CRAr;rEx XX.-The Fermi-Dirae Statistics and the Elec- 切。n(has. XXI.--The Models of Wave Mechanics一 XXII.--Applications of一the Free-Electron Theory PART V.----TRz BRILLOU N-ZON$ THEORY of METALS CHAFrim XXIII. :The Simple Theory of Brillouin Zones XXIV.---Insulators, Semi-Conductors, and Metals XXV. More Complete Zone Theories 142 1.K 1 151 165 174 179 179 192 208 PART VI.--ELECTso1gs, ATOMS, MuTA.L9, AND ALLOYS CIrAPTHR XXVI. The Alkali Metals XXVII.-Copper, Silver, and Gold XXVIII.--Some Metals of Higher Valency XXIX.--The Elements of the First Long Period XXX.--Some Magnetic Properties ArrENDIX.一NOTE ON THE ]EXACT FORMULATION OF THEUNCERTAIhi TY P'RINCIPLF OF丑EIRE?+TBEBC NAMR INDEX SUBJECT INDEX 213 213 226 241 `'58 274 284 286 287
INTRODUCTION. Ix the last 25 years very great advances have been made鱼the theory of the structure and properties of metals and alloys. In this branch of science some of the underlying principles have' been discovered era一 pirically, but most of the quantitative theory is due to the great progress in Mathematical Physics which has resulted from the developments of the new -quantum theory. Experience.has shown that students of metallurgy are greatly" interested in the new theoretical work, but are often quite unable to understand the papers or text-books written by mathematical physicists. This is due partly to the failure of the mathematician to appreciate the difficulties of the non-mathematical reader, and partly to the fact that most books assume a knowledge of subjects with which the metallurgical student is unfamiliar, although they are now included in the training of physicists, chemists, and crystal- lographers. This position is clearly unsatisfactory, and the present monograph has been written in the hope of providing a bridge by means of which the student of metallurgy may be led to an understanding of the general ideas on which the new theories are based. At the request of the Publication Committee of the Institute of Metals the monograph has been made reasonably self-contained, with the result that some of the subject-matter in Parts I, II, and III has already been treated in books of a reasonably elementary standard. It is hoped, however, that this repetition may be justified by the rearrangement of the material so as to form a general introduction to the theorv of metals. The book is written primarily for the .Honours t3tudent in metallurgy, and no attempt has been made to deal with mathematical technique. This method of approach means that equations have sometimes to be presented dogmatically, and it is hoped that this may be excused in view of the desirability of teaching the student not to be afraid of papers which contain numerous equations whose derivation he cannot hope to follow. In dealing with the applications of the theory it has been thought better to concentrate mainly’on the properties for which the theories have provided a reasonably straightforward explanation, and to avoid those whose interpretation is still in ' doubt. This means that the important property of plasticity has not been discussed, as it was felt‘ that this subject would have led too far from the electronic theory with which the book is mainly concerned.， The manuscript was prepared before the announcement of the 二 Vl1
VUx Introduction methods for the utilization of atomic energy, and as these have not yet any direct metallurgical application, it has been decided to omit them entirely from the present monograph. The author. must express his gratitude for the great help he has received from many friends in the preparation of this monograph· Particular thanks are due to Professor N.F.Mott, F.R.S.，and Dr一H. Frohlich, who have read the whole of the manuscript, to Dr. R. Sack, who gave much useful help, and to Dr. C. A. Coulson who has given great assistance with the proof corrections. Professor R. Peierls and 一r. H. Jones have also shown great kindness in discussing some of the theoretical points. On the metallurgical side, Dr. A，H. Cottrell, Dr. J.A. Wheeler, and Dr. H. J.Axon have given valuable advice as to the requirements of the University student, whilst Dr. G. V. Raynor has given much help in reconciling the points of view of the metallurgist and the physicist. The author must also acknowledge the great help given by Mr. N.F3.Vaughan (formerly Assistant Editor of the Institute of Metals) in connection with the preparation of the manuscript and diagrams. It need scarcely be said that the fact that the friends mentioned above have read the manuscript in no way commits them to the views expressed in the monograph. W.H一R. The Inorganic Chemistry Laboratory, The University Museum, Oxford. NOTE TO FIRST，REVISED，REPRINT A reprint of this Monograph has been required unexpectedly soon, and the original printing has therefore been retained with the addition of an Appendix dealing with the exact formulation of the Uncertainty Principle of Heisenberg. A few additions have been made to the suggestions for further reading, and some corrections and modifications have also been introduced.The author will be grateful if readers will noti行him of any errors or misprints which may be found in the text. 0功rd. W.H.-R. 6th一‘ebruary, 1947. NOTE TO SECOND，REVISED，REPRINT The Monograph is reprinted unchanged except for alterations on pp· 2:35--2:36 and 2-1-)3--2;")6, resulting from view work by Zener and Matyas. Oxford, W. H.一R. 22nd -November, 1947.
PART I.--THE GENERAL BACKGROUND. CHAFTEE I.--GENERAL IrrTRODUMON. TnE modern theory of atomic structure is the result of work in chemistry, physics, and mathematics, extending over more than 150 years, and involving very great changes in the underlying ideas. We do not propose to deal here in detail with the historical development of the subject, but it may be of interest to point out some of the more prominent stages by which the present position has been reached. The atomic theory of John Dalton provided the Science of Chemistry with a sound foundation, and its developments by the great chemists of the nineteenth century culminated in Mendeleev's Periodic Table of the Elements which appeared in 1869. This table, which is repro- duced in a modified form on p. 2, showed that a periodic repetition of physical and chemical properties was observed if the elements were arranged in a definite order which, with a very few exceptions, was the order of increasing atomic weight. These developments of what may be called the simple atomic theory of chemistry clearly supported the. view that matter consisted ultimately of atoms, which 运ordinary chemical reactions were indestructible, but which could combine together to form molecules in accordance with the principles of valency. At the same time the growth of the Science of Electro- chemistry, which began with the work of navy anal Faraday, suggested that electricity had some kind of an atomic structure, since the gram- molecular weights of the different ions were associated with small whole-number multiples of a definite electric charge. The periodicity re` ealed in Mendeleev's table suggested that the atoms themselves had structures, and that on passing down the Periodic Table, the changes in atomic structure involved the building up of a series of stable groups or units, so that each time a group was completed, the building up process began again.. The development of the Science of Chemistry led to the determina- tion of the relative weights of the atoms of the different elements, and these are now expressed on a scale in which the atomic weight of oxygen is taken to be 16.000. On this scale the ato而c weight of hydrogen is 1.0080, and the ato;uic weights at present accepted are given in Table I. In the middle and latter half of the nineteenth century, the develop- ment of the kinetic theory of gases showed that many properties of B 1
The‘ General Background the gaseous state could be explained on the assumption that a gLa consisted of an assembly of small particles, the average velocity of which increased with rise一of temperature. These“particles”were 且班e I2 尹、_ ，、‘ (Courtesy Clarendon Press. Periodic Table of the Elements. 吮 naturally identified with the molecules of the而stry, and several different lines of approach led to the conclusion that the number of molecules present in the gram-molecular weight of a substance was of
General Intro球u涵n the order number, 10U. Which Later and more perfect methods have shown that this is known as Avogadro's number, is equal to 6.02邓X1户. In the case of carbon monoxide, for example, the atomic w吨Its are carbon==12.010 and oxygen=16.000, and、consequently 28-(310 g. of carbon monoxide contain 6.0228 X 1023 molecules of CO. TABLE XI I.-International Atomic W吻Wt 1941. t.众一08ltI6008589八U 人Nl，.二0‘474、且片口1‘5 协DI.一.e、1，b场.sdt‘‘ra西公乃五.b卫吐D3.lgaraeblh 加加-肠州N(CN仓0P(P功K卜RRR凡R]RR18rscsesi为NSrs毛叭TIT]TI 人t. No. 人七。Wt. At. Wt. 686589。18655742144718632~010902全23遴勺旧旧LO 钊.刻卜侧U引钊积别钊截舍金倪么象卜fL别U大仗创9驯引创公争软2:5，334 TmsnTiwUv知YbYZnzr Beryllium Bismuth Boron Bromine Cadmium Csesiul n Calcium Carbon Cerium Chlorine Chromium Cobalt Copper 巧spzosium Erbium Europium Fluorine Gadolinium Gallium Germanium Gold Hafnium Helium Holmium Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum Lead. Lithium Lutecium Magnesium Manganese Mercurv Molybdenum. Neodvmium. 2697 121.76 39.944 74.91 13736 9.02 20900 1082 79.916 112.41 13291 4008 12.010 14013 35457 52.01 5894 6357 16246 167.2 1520 19.00 156.9 6972 7260 1972 1786 4003 16494 1-0080 11476 126.92 193.1 5585 83.7 138.92 20721 6.940 17499 24-32 5493 200.61 95.95 144.27 Neon Nickel Niobium (Columbium)， Nitrogen Osmium Oxygen。 Palladium Phosphorus Platinum Potassium Praseodymium Protoaotinium Radium Radon Rhenium Rhodium Rubidium Ruthenium Samarium Scandium Selenium Silicon Silver Sodium. Strontium Sulphur. Tantalum Tellurium Terbium Thallium Thorium Thulium Tin Titanium Tungsten Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium 20-183 5g-69 92.91 14·的9 190.2 16.0000 106.7 30.98 195.23 39096 140.92 231 22605 222 18631 102.91 8548 1017 15043 45.10 78.96 28心6 107.880 22997 8763 32心6 180.88 12761 1592 20439 232.12 1694 11870 4790 18392 238.07 5095 1313 173心4 8892 6538 91.22 318384355850687479683941292271仑3t，6弓72312石乃2心 1513口.5:3宙丘勿|导r多22肠eel吞33宁了:e45723·58712846 姗一Al8bAAsBaBeBiBBrCdCsCaQCeclCrCoCu、ErEu， GdGaGeAuHfHeHoHinllr几KrLaPbLILuMghlnHgMoNd
4 The General Background The determination of Avogadro's number led to an estimation of the actual as distinct from the relative weigats of the atoms of the different elements, and this knowledge,- when combined with the known densities of substances, suggested that the distances between atoms in solids were of the order 10-7-10-8 cm. The subsequent development (1913-20) of crystal analysis by X-ray diffraction. methods confirmed this experimentally, and in so far as the interatomic distances in crystals are a measure of the sizes of the atoms concerned; we now know that the values lie between 10-7 and 1Q--8 cm., and the interatomic distances in solid metals have been determined to a degree of accuracy which is often as high as I part in 50,000. The periodicity，shown in Mendeleev's table suggested clearly that atoms were not analogous to very minute billiard. balls, but that they possessed definite inner structures. The modern theory of atomic structure is founded essentially on the experimental work at the end of the nineteenth century which led to the discovery of the electron. In this work it was shown that under appropriate conditions, such as electrical discharges in gases, high temperatures (ther面onlc emission)，and strong electric fields, many substances could be made to emit“rays”which on passing through electric or magnetic, fields behaved as though they consisted of very minute particles car示ng charges of negative electricity. These experiments showed that although the velocities of the emitted electrons varied greatly with the experi- mental conditions, the mass m and charge e of the electrons were always the same, irrespective of the substance from which the electrons were produced. The values accepted at present for these constants are: electronic mass, m==91066 x 10-28 g. electronic charge, e==4.8025 X 10-10 electrostatic units. The mass of the electron is thus aboutl矗万 that of a hydrogen. atom, whilst its electric charge is the same as that on a univalent negative ion.* The development of the methods of mass spectroscopy? by J. J. Thomson and F. W. Aston enabled the masses and charges of electrons and ions to be studied in detail, and th?:work showed that whilst both positive and negative ions existed; only negative electricity was associated with a mass as small as that of an electron.t This *The charge faraday, and is on one gram-molecular weight of ual to 96,487 absolute coulombs. is therefore eq 90 487令60228 x 1013 ==1.602iv3 units, the X a univalent ion. is called, a The charge_ on one single ion 10.19 coulombs. SMce 1 coulomb二2.99776 x 109 electrostatic charge on one ion 1.50203 x 299776 x 10-10==4.8025 x 10-x0 e.s.u., which for the electronic charge. is the value gi t We are the discovery metallurgy. here ignoring the latest developments of positive electrons, since this has of at Payslcs present which have led to no application to
General Introduction 5 led naturally to the idea of an atom as consisting of a relatively heavy, positively charged body to which was attached a number of light, negatively charged electrons, and from this it was an easy step to the concept of a nuclear atom with negatively charged electrons revolving round a positively charged core. This concept of a nuclear atom received direct experimental confirmation from the investigations of Rutherford on the deflection of二一particles*when passing through matter, and his work led to the conclusion that an atom consisted of a minute positively charged nucleus, surrounded by a sufficient number of electrons to keep the atom as a whole neutral. The dimensions of the nucleus are of the order of l0-1$ cm.，and acre thus very small compared with the distances between atoms in molecules or crystals (10-?--10-8 cm.), though the positively charged. nucleus contains almost the whole of the mass of the atom. The zone occupied by the surround- ing electrons is‘ of the order of 10--7-10-$ cm.，and is thus of the same ma脚tude as the interatomic distances in crystals. The positive charge on the nucleus is equal to+Ze, where Z, the so一called atomic number, gives the position of the element in the Periodic Table. Since， the atom as a whole is neutral, an atom of atomic number Z is sur- rounded by Z electrons, so that an atom of, say, copper of atomic number 29 contains a nucleus of charge+29e, surrounded by 29 electrons. The development of the idea of a nuclear atom was accompanied by increasing evidence (1910-1930) that under some conditions the nuclei of atoms were not indestructible. The first discoveries were in connection with the phenomenon of radioactivity, where it was found that some of the heaviest elements underwent spontaneous disintegration into new elements, the process being accompanied by emission of positively charged二一particles, negatively charged electrons (p一rays), or very penetrating radiation (y-rays).These investigations showed that certain elements, notably lead, could exist in forms which·had different atomic weights, but were chemically indistin扣shable. These discoveries, in the hands of Soddy, led to the concept of isotopes, this name be ug given to eases where an element could be shown to exist in the form of atoms which were of different atomic weights, but which had the same positive nuclear charge, so that they occupied the same position in the Periodic Table and were chemically in distinguighable.t Subsequent work showed that nearly all the elements met with on the *。一Particles or。一rays are are doubly 卞Some slightly on the weights 蟹糕老some radioactivee transformationsatoms which have lost two electro黔 s (e.g., ratesof the atom养reaction involving difuut the differences are us怒may,very
6 The General Background earth are really mixtures of isotopes;thus chlorine, of atomic weight 35.457，consists of a mixture of isotopes of atomic weights 35 and 37. The conclusion reached was, therefore, that the characteristics of the atoms of elements are determined not by their atomic weights, but by the positive charges on their nuclei. The order of the elements in the Periodic Table is thus the order of increasing positive charge on the nucleus, each step in the table corresponding to an increased nuclear charge of+e. This order was first established conclusively by the X-ray spectroscopic investigations of Moseley (1912)，whose work showed that in the few cases where the order of the.elements in the Periodic Table was not the order of increasing atomic w吨ht, the regular increase运nuclear charge was maintained, Thus, ,as can be seen from the table on p. 2，the order - of the elements , in the middle of the First Long Period is iron一一)cobalt~一爷 nickel，for which the nuclear charges are+26e,+27e, and+2$e, respectively, whilst the atomic weights are 55-$5, 5894, and 58.69，so that cobalt and nickel are out of order as regards atomic weights, but in order as regards the nuclear charges. Later experimental work on atomic structure has been concerned mainly with the structure of the nucleus, and although these develop- meats lie outside the scope of the present book, they are of the greatest possible interest,* and have led to methods for the娜ificial breaking up of atomic nuclei. From what has been said abote, it will be ap- preciated that the characteristics of an element depend essentially on the atomic number, i.e.，the number of positive charges on the nucleus, and it may be said that for the present the structure of the nucleus is in general unimportant for the study of metals and alloys, although the reader should note that the artificial production of radio- active isotopes of the more common elements has led to坛teresting methods for the study of such subjects as self-diffusion. In this work minute quantities of a radioactive isotope时an element such as silver are prepared and electroplated on to a piece of silver or of a silver alloy, and after an appropriate heat-treatment, the penetration of the radio- active isdtope can be studied by measurements of the radioactivity of the metal at different depths. From the above survey it will be evident that the general concept of a nuclear atom is the result of a long series of experiments of a widely varying nature，which indicate that the theory of atomic structure must involve the behaviour of electrons in the fielcb of force of a minute positively charged nucleus, and at distances from the nucleus which do .尸节叭。，。。J八.一。，.。，~，1吞。L‘，1，二。L，一“T~‘.‘J。.“:~一人二几L~一:~nL一_三‘~” 一’1’ne reaQer may consult a匕ook sucn as”In七roductiontoAtomic犷hvaios," D。·-1'O1SIiSKy. LOIIQOII:1b4`7. ( LOngnlans, green的dVo,}
General Introduction 7 not excelJ about 10-7 cm.Before describing the theory of the subject, we may consider briefly the units of measurement which are usually employe4in connection with the structure of the atom. The unit of length in most general use for atomic phenomena is the Angstrom unit- (A.)，which is equal to 10.8 em. In X-ray crystal- lography an unfortunate position has grown up in which lattice spacings are given in what are usually called Angstrom units, though they acre not exactly equal to 10--8 cm. These“crystal Angstroms”are actually based on the Siegbahn scale of X-units, an X-unit being defined so that the (200) spacing of calcite‘ at 180 C. is equal to 3429.45 X-units. This definition made an X-unit as nearly as possible equal to 1(}3 A. on the basis of the then known value of Avogadro's number. More accurate work has shown that a slight error was made, and the crystal- lographer's Angstrom unit requires multiplying by 1.00202 in order to convert it into the true Angstrom unit of 10-8 cm. The theory of atomic structure is not yet sufficiently advanced for this difference to be significant in any but a small minority of cases;nevertheless, the reader should bear it in mind and note that proposals have been made to use expressions such as“absolute Angstroms”to denote true Angstrom units of 10-8 cm.，and expressions such as kX. or“crystal Angstroms”to denote the crystallographic unit which is equal to one thousand Siegbahn X-units. For some purposes it is convenient to use the so-called“atomic unit of length,'’which is equal to 05292 A. As we shall see later (p. 14), the Bohr theory assumed that the one electron of the hydrogen atom could travel round the nucleus in a number of definite orbits or g ationary states, and the atomic unit of len妙h is the radius of the first Bohr orbit, and is equal to: h2 44me$' where m and‘are, respectively, the mass and charge of the electron, and h is Planck's Constant of Action, which is referred to later and which equals 6.624 X 10-27 erg-second: The atomic unit of length is em川oyed because its use avoids the necessity for working out terms of the form 0/44w,2 which occur in some calculations. For一the measurement of energy, the results can be expre&4ed either as the energy change per atom or as the energy change per grain atomic weight. When we are concerned with the individua' atom or electron， the most convenient unit of energy is often the energy change involved wken one electron falls through a potential difference of 1 volt. This quantity of energy is called an electron volt, and is equal to 1.60203 X
8 The General Background 10-12 erg. Avogadro's number states that there are 6.0228 X 1023 atoms恤’the gram atomic weight of an element, and hence an energy change of one electron volt per atom ifs equivalent to 1.60203 X 10-1$ X 6.0228 X 10P二9.6487 X 1011 ergs per gram atomic weight. Now I joule is equal to 107 ergs, and 1 caclorie is equal to 4.1855 joules. Con- sequently an energy change of one electron volt per atom is equivalent to: 9.6487 X 1011 不1855 X互07“ 23.05 X 103 cal. per g.-atom 23.05吨-cal. per g.-atom. If, therefore, an electron in an atom or a molecule undergoes an energy change of 1 electron volt, the energy change involved is equivalent to that of a heat of transformation or reaction of 23.05 kg.-cal. per g,- atom, and this conversion factor should be borne in mind when trying to visualize the magnitudes involved when results are expressed in electron volts per atom. The latent heats of fusion of most metals lie in the range 0.5--10.0纯.-cal. per gram atomic weight, so that a change of卉of a n electron volt per atom is of the same order as a latent heat of fusion· The energy changes involved in atomic transformations are also sometimes expressed in terms of wave numbers or frequencies. It is well known that the phenomena of interference effects led to the development of the wave theory of light, but that when absorption or emission processes occur, the facts can only be explained by means of the quantum theory. This theory was first advanced妙Planck (1901)in order to explain the distribution of energy among the different wave-lengths of radiation emitted by hot bodies. According to the quantum theory (see p. 23)，energy in the form of radiation can only be emitted in units or quanta of magnitude E == hv, where E is the energy, v the frequency of the radiation, and h is Planck's constant referred to above.人beam of radiation of frequency v is thus the result of an immense number of atomic processes each involving an energy change equal to hv. Conversely, if an atom gives rise to emission, as the result Of an energy, change E, it expels one quantum of radiation of frequency i lh. It can therefore be understood that in many cases where atomic processes result in the emission or absorption of radiation, it is convenient ,.,o express energy in terms of frequency by means of the relation E二11. The velocity of light is -usually denoted by the symbol c and is equal to 2-99775 X 1010 cm. per second, and since the velocity is equal to the product of the wave-length and the frequency, it follows that Ii睁t of frequency v, has a wave-length a equal to c Iv, whilst the uuve number, n,
Geneyal I ntyoduction 9 or number of waves per cm.* is equal to v Jc. If, therefore, an atom undergoes a transition血which it emits one quantum of radiation of frequency v, the energy change involved is 6.624 X 10--27 v ergs, and since the wave number，equals v/c, the energy of a quantum of radiation of wave number n is 6.624 X 10-27 X 2.99776 X 1010 X n=1.986 X 10-16 X n ergs per atom. It is possible to use ,wave numbers as a measure of energy, and combination of the above factors will show that an energy change of y electron volts per atom is equivalent to a wave number n二8.0665 X 102 X y, and an energy change of wave number ，is equivalent to 1.2397 X 10-4 X，electron volts per atom. The wave- lengths of visible light are of the order of 30007000 A.，so that an atomic transition which乡ves rise to visible light involves an. energy change of the order of 30,000-15,000 cm，一I wave numbers or ca. 3.5-1.8 electron volts per atom. The wave-lengths of X--raps are of the order of I A.， so that the atomic transitions involve energy changes about 1000 times as great as those which梦ve rise to optical spectra. 狗.1，which is taken from H. w. Thompson's“A Course of Chemical Spectroscopy”(Cambridge University Press) enables a comparison to be made between the above different units of energy over a certain range. The reader will also encounter energies measured in Rydberg Units, where I Rydberg二13.59 ex. This quantity of energy is equal to the first ionization potential of hydrogen, i.e.，the work required to remove an electron from a hydrogen atom in its normal or lowest energy state. It is also useful to remember that饭the equation for a perfect gas, PV二Re, the quantity Re represents the energy of a gram-molecule of the gas at the absolute temperature e;the constant R bas the value 1.98647 cal. deg一1. The corresponding equation referred to a single molecule is pv=ke, where k is the so-called Boltzmann's constant, and equals 1.38047 X 10-1" erg deg一1. At room temperature 8 is equal to approximately 300, so that k8 is of the order 4 X 10`14 ergs, i.e.，of the order of 3 X 10-2 e.v. This should be remembered when equations are 名 encountered with terms of the form。一Fe, where E is the energy asso- ciated with some process. 'Vote on尸hysical Constants. _.In吟last 10 years slight啊nges have。 occurred11热the accepted values7 73一‘any of the physical constants usect in atomic， theory. -hrtese changes，result rrom LAOr discovery ot errors in the oil-strop method of aetermming the electronic charge, e， .It has, unfortunately, become a common practice to use the symbol，加t杠 for the frequency and for一。wave number which，have denoted by the sym如1 n, and the reader must be careful to distinguish between c abee where，means the real frequency and those where it means the wave number.
10 The General Background 10,000 I Ovo 00 r.}9p0 30 8,000 71000 40 1 5,000 6,000 2·0 50607080 20,000 25p00 5,000 4,000 5·0 30,000 31000 —90 4·0 1001101z0130140150160170180190z00川240260 40,000 5如00 2,000， }.j6叩00 -x0,000 11500 80,000一 90,000一 1o0,000 气em.-I. 恒二卫so一一上 kg.-Cal./g..mol. 【口。公如梦Ca;*ridge 】1 for legend.) 5·0 6·0 7.0 8·0 9·},}0 10·0 11·0! } I之·r`} 0.丫。 TJniversfty Prcu. FIG. 1.(Bee p.
，lre.玛d rLal到le ot)l eTU ，b钾n‘ General Introduction and also in some methods of de taken from the paper by R T 。rmining the ratiorm Birge. * The posi hle. The values given tion at the moment of rather papers, in the correct confusing, although because 二口.. B in lg e's values now being used in most they have not been .Uesare officially majority calculate of teat-books.玩the d values found in publis h re习en recognized，_ an11 V L 0008 IT, IIas d are not to be found not been possible to ed papers and based on the older values of the physical constants. The changes are sometimes considerable:thus value of Avogadro's number was 6.064 X 1023, as compared with theof 5.0228 x 1023. Many of the constants are determined by methodsa knowledge of some of the other constants, and the reader who wishesthe details should consult the papers by Birge. present value which involve to understand Velocity of Light, c 二2-99776 X 1010 cm. sec一1. 1 Faraday Avogadro' ，F二95,4$8 absolute coulombs g.-equiv.-I s Number. 1Y二6.022-8 x 1023. Electronic Charge,。二4.8025 x 10-10 abs. e.s.u. Planck's Constant, h,二6624 Electronic Boltzmanl Gas Const Energy in Energy in molecca Mass, m二9.1066 X 10-27 erg sec. x 10-28 9- 9)., k二1.38047 ale, R二1-98647 x 10-1$ erg deg.-t， cal. deg volt electron( gram-molecule 052 kg.-cal. see p. 8 )“ 一1 1.00243 for one absolute x olt 10-1s ergs. electron per Magnetic Moment of 1 Bohr Magpoton二09273 x Wave-length associated with one absolute volt (see 10--20 p. 9) erg gauss-1. 二12395 x 10-$ cm. abs. volt. Wave Number associated with one absolute volt-'. 1 Crystal Angstrom节1 kX Atomic Unit of Length,。= .unit -- 0.5292 x absolute volt (see p. 9), n = 8067 cm。一1 1-00202 x 10-9 cm. 10~$ cm. *.Rev. Modern Physics, 1941,r, YIN，移， ap户浓路in”reports on rrogress }n London;. 233;another paper Physics," 1941, vol. by S, the same author {P妙sical Society, FiG. 1.---In this figure Thenext we have e scale is the right-hand scale is a the corres-pondmg scale logarithmic scale of electron volts. xplained that an energy ofkg一 change of I will be seen e，V。 23.05 kg.- cal./g scale of Fig. I .-molecule, and it gal.怜-molecule.。即P.，S 1moiecule is equivaient to that 1.0 e.v. on corresponds with 230.5 kg.-cal./g.-molecule the right-hand on the scale. When an atom or molecule emits a quantum of radiation adjacent v, the change in energy, E, of the atom is given by the relation: of frequency E“by = hcl A where A is the -rave-lengththe right shows the wave-1粼c is the velocity of light.p The third scale fromhs of the radiation corresponding to the energy changes of the first scale of ex./molecule or of the second scale of kg.-cal./g一 molecule. Visible light has wave-lengths of the order of 3000-7000 .A.，and hence corresponds with energy changes of the order of 4-2 e.v.。 The left-handscale is the corresponding scale of wave numbers, i.e., number of waves/cm.
CHAPTER II.---THE FIRST THEoRY。，THE HYDROGEN ATom. IN developing the theory of atomic structure, attention was naturally concentrated first on the hydrogen atom, since this consists of a single electron in motion round a nucleus with a single positive charge of +。.In its simplest form, the first theory of the hydrogen atom assumed that the electron described a circular orbit round the nucleus, so that the inward attraction of the nucleus was balanced by the outward centrifugal .force. Although the idea of a precise mechanical orbit has now been completely discarded, many of the general conclusions of this model are still valid and must be appreciated. An electron describing a circular orbit round a positively charged nucleus of charge+e, is attracted妙a force equal to e2/r2- The electron possesses both potential and kinetic energies, and the poten- tial energy, TY, must be measured relatively to some arbitrary zero. Far this purpose the zero of potential energy is assumed to be that of an electron at rest at an infinitely large distance from the nucleus. In a circular orbit of radius r, the potential energy is then negative, and equals一e2介.*The kinetic energy equals扣u2, where“ is the velocity. Since the centrifugal force, mu2lr, is to be balanced 妙the attraction, we have: m沪 e2 下-==rs I (I) so that the kinetic energy, K, is given by: ，户 八“#mu&==玄r·’二’且tLl and the total energy, +., is therefore: E二potential enei gy+kinetic energy }2 02 -V一卜丢乙二 r’.r 。2 ，气沙Y I} I-.之l —酋.丁·…i气。) 一r The total energy is thus always negative t and becomes increasingly *The negative sign of the potential en. and should valley, and give rise to no difficulty. If, assume the potential energy ergy is the result of }he choice of zero for example, we consider a hill and a f a stone in the valley to be zero. its potential energy on the top of the hill clearly arbitrary, and we mig of the hill zero, and in this case t In some treatments it is oust negative total energy, causes con允咸。〔as tea 80 that Y二+ positive. But this procedure is the potential energy on the top y in the valley would be negative. ;e some symbol, sayr，for the is this convention which often whether large or small orbits have the greater total 12
The First ?'henry of the Hydrogen Atom 13 negative as the orbit becomes smaller. It is thus necessary to add energy if the electron is to be pushed out into a larger orbit. The kinetic energy equals jet/r, and hence the kinetic energy of an electron becomes smaller as the orbit becomes larger, and on comparing a large and a small orbit, vie see that the larger orbit has the greater total energy and the greater potential energy, but the smaller kinetic energy. In so far as it is justifiable to think of“orbits”(see p. 44), these considerations apply to all the later theories.
CHAPTER III. THE THEORY of BOHR. THE defect of the above classical theory was that it did not explain either the reason for the formation of sharp spectral lines, or for the stability of the atom.An electron moving in an orbit as described above would, on the basis of classical mechanics, continually approach the nucleus, with a steady emission of radiation of gradually chap咖g frequency. The first step towards the solution of this problem was made by Bohr, whose )ri脚al quantum theory of the atom involves two main assumptions: (1)The electrons are assumed to revolve round the nucleus in definite orbits called Stationary States, and emission or absorption of radiation takes place by an electron jumping from one stationary state to another. When the一electron jumps from an orbit of energy凡to one of energy 凡，the atom emits or absorbs a quantum of radiation of frequency v, given by the Einstein relation: 凡一E:二hv,..…I (4) where h is the fundamental constant of Planck and equals 6.624 X 1尹7 erg-second.Emission of radiation takes 凡，and absorption occurs if the atom gains place if凡is greater than energy during the transition. (2) The only stationary states which are stable are assumed to be those for which the angular momentum 1吕an integral multiple of h /2-m. numb In this way, an electronic orbit or state is associated with a whole er, the so-called revolve in an orbit for quantum number. which the angular An electron can, for example, momentu h 2h 3h m is添，丽，派” or in general吵。where，is the quantum number。:the orbit. In。 一G六~ simple circular orbit, the angular momentum equals mur, so that the condition for quantization may, be written: nh，，，、 Mur“n·……1气a) G兀 This condition is一 an arbitrary' postulate of the theory, and when com- bined with the relations given above leads to the expression: ，24w41丫 if
% B二一一节r一X:1.·…1 <o) 几甲妈. The negative energies of the orbits are thus proportional to the inverse squares of whole numbers. From equation I (4)，the frequency of the 14
5 11 The Theory of Bohr quantum of energy emitted when an electron jumps from an orbit of energy El to one of energy E2 is equal to (E;一E,) /h. Since the wave number is equal to v/c, where c is the velocity of掩ht, the wave number associated with a transition from an orbit of quantum number、to one of quantum number n2 will be: 2沪mtA, 'CM ( 1nit一1)n$2/一“( nis 生、 ”。，/ .r I (7) where ::二 2n chs This agrees with the fact that some of the fines of the hydrogen spectrum can be classified in series whose wave numbers involve the differences of the reciprocals of the squares of whole numbers. The arbitrary assump- tions of the theory were, of course, chosen to explain this fact, and the really striking point about the theory is not so much that it predicts the above types of series, as that the numerical value of the Rydberg Con- stant, R, is almost exactly that required by experiment, and an exact agreement is obtained when the theory is elaborated so that the electron is regarded as revolving not with the nucleus as the centre of revolution, but about the centre of gravity of the nucleus and the electron together, this being, of course, the more correct assumption. The circular orbit is the simplest example of the more general type of orbital motion, which is that of an ellipse with the nucleus at its focus. The Bohr theory was extended to the case of elliptical orbits with the additional complication of the， variation of the mass of the electron, in accordance with the Theory of Relativity.* In the original theory of elliptical orbits, each electron state is characterized by two quantum numbers n and k. The principal quantum number,，，is a measure of the total energy of the orbit, the negative energy varying inversely as n2 in hydrogen-like atoms, whilst the major axis of the ellipse is also proportional to n2, so that the orbits become larger with increasing”.The secondary quantum number, k, is a measure of the angular momentum of the orbit, which is equal to kh/27r. The ratio k/n gives the ratio of the minor to the major axis of the ellipse, and *According to the at rest is equal to m,O, where of light. The mess,，，of a pa y of Relativity, the _ energy of -a material mo is the rest mass of the particle and c the rticle moving with veloc柳。，is乡ven by the Theory of Relativity, .particle 盆cityrelation ，’饰IVI一吞，so that the mass varies with the ve: appreciable at very high- velocities. The_ total energy velocity“is equal to“.+V, where V is-the potentia compared with c, the total energy is equ丝to moo士 called the mass energy of the particle, and女mouy- is the particle moving with energy, and if，‘is small .+V. Here协.c $ is io energy.
16 ?'he General Background thus determines its eccentricity;k can have any value from 1 to，， and in any one hyd: oxen-like atom all orbits with a given value of， have the，major ass of the ellipse. The orbit for which k=I is the moat ecxxentric of those with a given value of，，-and is hence the obit in which the electron travels farthest from the nucleus, whilst the orbits for which n==k are circular. In the elliptical obits, the velocity and hence the mass of the electron vary with its distance from the nucleus, and the effect of this is to produce a precession of the orbit as shown m Fig. 2。This precession is very slow’com- pared with .-he time of a complete revolution in any one orbit, and in \\\ //1 / !!!.了了了/r / 一 ﹄、、 ﹄\1 〔cawUsy G..BeU and ion. FTc . '"'.一一。illustrate the prep-egsioa of an elliptical orbit. the orbits for which the e-ffect is greatest, the electron travels round the ellipse about 4-0,000 times for each revolution of the major axis. This refinement of the theory was due to Sommerfeld, anal it led to a fairy sat'fa ctorv tbeory of the hydrogen atom. although it was necessary to 运行d舫肠third quanuUm number认order to explain some of the facts. T I。.f:he。叮also Che -}--:-.heorvr also led to an explain-ition e .'f the spectra of the positive ions .1-10- and Li++，which eonsist of一one electron revolving round nI-1Ul*-1J;. A- f, -charge+2e an-a'+3s, respectively一Some progress was also nt.ad。in比。interpretation of the spectra. of more complex atoms, ? a _龙1盯妙比佣e，双。h as the alkalis, whose spectra slowed. resemblances “。rt3Cti'r。fL , drogr-U‘T}.fi alkali metals arc univalent. and their free
The Theory of Bohy。17 atoms contain one relatively loosely bcund electron (see p. 60)，which was regarded as revolving round the“core”formed by fiche nucleus and the inner electrons. .A. distinction was made between“non-p血e- trating”orbits, in which the electron remained outside the core during the whole of its trajectory, and“penetrating orbits”in which the electron, when nearest to the nucleus, plunged into the region in which the electrons of the core were moving. In this region the attraction of the nucleus was no longer of the inverse square type., and the effect of this was to give a motion analogous to that of Fig. 2，but with a much more rapid revolution of the axis of the ellipse, so that there were only a few branches of the rosette to each complete revolution of the major axis of the ellipse. In spite of its great success, the Rohr theory was not completely satisfactory, for还the first place facts were found with which it-did not agree，and, secondly, it involved a logical inconsistency, in that when an electron was moving in an orbit, it was assumed to obey the laws of classical mechanics; though.these were subjected to the apparently quite arbitrary restrictions of the quantum hypothesis. The solution 、to this difficulty was found, not by making further arbitrary assumptions, but by recasting the laws of mechanics into a new mathematical frame- work, and we may approach this difficult subject by considering first some points in connection -with the Uncertainty Principle of Iieisenberg. 8 QOGESTIONS FOR For a detailed account of the Rohr th READINIT0. reader ，J、.f产写J， 峪‘上1争.山 人ndrade FURTHRR eory, the N.da C.“The Structure of the Atom." ,lid Son may consult: London:1527 (G. Bell Sidgvick, Pres幻. V.“The Electronic Theory of Valency." Qgfford:1527 (Clarendon
CHAPTER IV.---THE UNcERTAIWY PRINCIPLE OF .tlEISENBERG. IF the previous pages are examined critically, it involve the assumption that quantities such will be seen that they “position, velocity, momentum, &c.，may be ascribed to an electron at anv instant. and that -.口，一 these quantities havet h eal meanings amociated with them恤the 冷if 鑫七.j classical mechanics.- 比e physi From the modern viewpoint, it is mea speak of a physical quantity unless we can wingless describe a method which not yet actually realizable, will at least in principle enable us to measure the quantity concerned.、If measuring quantities such as we consider the ordinary position, velocity, &c.，it that they all involve the assumption that we can alteration produced measure the length the act of measurement. body, we must compare ignore most of the If, for example, we it with a standard scale. making contact between one body anal another, If this is done we are uncertain as to the exact deformation produced. by the actual contac七. If we seek to overcome this difficulty by light to compare exerts a pressure, rising a beam of the object with the scale, the indicating beam of light and we are again uncertain as to the exact deformation caused. With.ordinary objects these effects are negligible, but when we come to consider electrons, different. so small as to be the position is very The usual illustration of the Heisenberg Principle is that of an attempt to inea名Ure the simultaneous position and velocity of an electron. To do this we might in principle beam of light, and so observe the positions of time and hence deduce its velocity. microscope is limited, and is given by the *1tOA }"TlTA m7n}.}r}nr},n}r.。，J。 UAG v”.1111 VI VotrvFca fUU I* of the electron at two instants The resolving power of a relation Aa，二 人 sin 8' where ), is the wave-length of the light employed, and 28 the aperture of the objective. The accurate measurement of position therefore involves the use of light of a very short wave-length and of are objective of high aperture, the objective receiv'_-Lg 1妙t from all directions included within the angle 28. The一work of Compton has, however, shown that when吨ht interacts with an electron, the latter undergoes a recoil, the process being known.the.Compton Effect. This process is a tppioal quantum phenomenon in which a ray of 1妙t of frequency v behaves as though it consisted of a stream of particles or砂otons eaeb of energy ，The experiment is a purely hypothetical one. is
The Uncertainty Principle of Heisenberg 19 AV, where h is Planck's requires interaction with imagine the photon to through an angle二，the constant. The observation of the electron at least one photon, and in this case if we travel along the x-axis anal to, be electron acquires a momentum whose oom- ponent light. in the x direction (1 --- cps a), Ohere c is the velocity of hv一c .1S The observation of the first position of the electron will alter the velocity of the electron妙an amount which -increases with ，(i.e.，as the wave-length becomes smaller) and which cannot be estimated exactly, because the objective of high aperture receives light from the whole range of the angle 28.We see, therefore, that the very factors which favour the accurate measurement of the position produce the greatest uncertainty in the velocity and trice.，”。一No method can be devised which avoids this difficulty, and the Uncertainty Principle of Heisenberg states that if Axx is the uncertainty in the position and Ap the uncertainty in the momentum，then the product OxAp has a minimum value of h, where h is Planck's Con.at nt of Action. This principle is not一confined to the above case. If, for example, we try to measure the energy, E, of a particle at a particular instant of time, t, we find that the conditions necessary to produce the greatest accuracy for the energy are: those which produce the greatest uncertainty in the time, and again AE AM -t h. This implies that if we are dealing with an electron in a stationary state the latter will only have.an exact energy if the time is completely indeterminate. If we deal with a restricted period of time, then there will be a slight uncertainty in the energy. With periods of time of the order of one second, the effect is negligible, because of the small value of the constant h, and it is for this reason that we usually obtain sharp spectral lines, whose frequencies are connected by the relation E:一E2二by In some cases an atom may go through a series of changes, in which it lasses through a number of energy states E,，EVES。二In such a case it may happen that one of the states has a very brief existence, and if so its energy is s纯htly uncertain，in accordance with the Heisenberg Principle, and a diffuse spectral line results.* This effect has, in fact, to be considered in the study of the soft X-ray spectra of solids, which has thrown great light on the nature of metals. Similarly, if we try to measure the angular momentum, M, of a particle revolving round a fixed point, and the angle ca made with some fixed reference lin$ by the line joining the particle to the fixed point, the uncertainties are given by *There are, of course, many other factors which。砂produce diffuse spectral Unee.
20 The General Background AM)co -q-,h. It will be noted that in all the above cases, the product of the two quantities connected行the Uncertainty Principle has the dimensions*ML2T-1, i.e.，the dimensions of action (see p. 27).It will be noted that the Principle refers to the product of the two uncertain- ties and refers only to the minimum value of this product. In considering the meaning of the Uncertainty Principle we may adopt two points of view. On the one hand, we may argue that our methods of measurement are at fault, and that if these could be per- fected we could measure the real momentum-and position at the same time. This clearly leads to difficulties about the meaning of reality, since we are practically compelled to say that by the“real position and momentum of an electron at a given instant”we paean something which, by the very nature of things, we cannot measure. Apart from this we should still be left with the problem of how the electron would behave还the hypothetical world where momentum and position, energy and time, &c.，could be measured simultaneously. Without going into details, it may be said that this line of approach has failed. The alternative is to accept the Uncertainty Principle as embodying a real characteristic of the physical world, which had escaped our notice in large-scale phenomena, because the magnitudes involved are very much larger than Planck's constant h, but which is of supreme importance in connection with the behaviour of particles of such small mass as an electron.It is from this point of view that such great progress has been made，though examination will show that a very strange world is being visualized. Suppose, for example, that we consider an electron contained in a one-dimensional box of len琳h L. appreciable when the length L is of atomic dimensions. Heisenberg's Principle then states that the uncertainty in momentum must be a t least AIL, and we cannot consider changes of momentum of less than this, since to do so would imply that the uncertainty in momentum was leas than h/L, and the electron. would therefore not be confined to the length L. These considers b ions clearly lead to great ditfictuities-if we try to describe small-scale events in terms of our usual ideas ,3f space sad time, since we have become so accustomed to imagining that each. of the *Position has dimensions multiplied by position hws two quantities which are the same instant. L. and mass万，and vel dimensions ML=少一1. canonically conjugate L厂T，。momentu The technical expression is th m at cannot he measured exactly at
The Uncertainty Pyinciple of Heisenbeyg 21 properties, position，velocity, kinetic energy, momentum，time, &c.， can be defined exactly, without reference to the others, that we find it difficult to form a mental image of the restrictions imposed by the Uncer- tainty Principle. In the methods developed by Heisenberg and Dirac, all attempts to describe phenomena in terms of our usual concepts of space and time are given up, and are replaced by systems of mathe- matics, which deal only with the quantities we actually observe, and express these in suitable mathematical forms. Although quantitatively satisfactory, these.methods are difficult to visualize, and for the non- mathematician it is more easy to consider the alternative line of approach, the so一called wave mechanics of Schrodinger, since this expresses some of the results in a way to which we are already accustomed. The methods of Dirac, Heisenberg, and Schrodinger are mathematically equivalent, as of course they must be of es-ch is to lead to the correct answer, but some of the results of wave mechanict can be visualized, although care is necessary to avoid using the wave analogy unjustifiably. }Y ate:It has been pointed out that confusion may arise from the fact that ，。。Qn耘，this is discussed on n_ 2R4_
CHAPTER V. THE IDEAS o8 WAVE MECHAMCS. IN the preceding sections we have described the general failure of ordinary mechanics to deal with problems of atomic structure, and we have seen how Heisenberg's Principle suggests that when.，Con- eider events on the atomic and electronic scale we can no longer regard properties such as position, time, velocity, momentum, kinetic energy, &c.，as being defined exactly and independently without reference to one another. If we ignore the historical development of the subject。 and consider the experimental evidence alone, we rosy say that the solution to these difficulties lies in the fact that electrons have been shown to沙e rise to diffraction and interference effects. This dis- covery was made first妙I)avisson and Germer (1927), and by G. P. Thomson(1927), who showed that diffraction patterns could be observed and recorded photographically when beams of electrons were reflected from crystals. This remarkable discovery showed that electrons pos- sessed the properties of waves as well as of particles, and a beam of electrons of velocity u was found to behave as though it were associated with waves, of wave-length X,廖ven 入二 by the relation h 歹， where h i$ Planck's constant, and p = momentum. For electrons whose velocity is much smaller than the velocity of light, c, the momentum p may be put equal to mu, and the relation reduces to: h 入:二二—。 r洛u For abbreviation we shall always use this simplified form, and this relation is the basis of the methods of investigating crystal structure by electron-diffraction technique;considered experimenta衍it is the foundation of the whole of the modern theory of atomic structure and of the properties of metals. We see, therefore, that the electron which was at first thought to possess only the properties of a charged particle, exhibits also characteristics which are usually, associated with waves, and it is well known that light shows a }r dual character. In this respect the development of optical theory may be divided into three well- marked stages. The older science of Geometrical Optics dealt with the passage of rays of light from one point to another, and led to the dis- 22
The Ideas of Wave Mechanics 23 covery of the laws of reflection and refraction. The fact .that light rays travel in straight lines in a uniform medium naturally suggested that吨ht consisted of streams of particles or corpuscles, and a cor- puscular theory of light was advanced by Sir Isaac Newton. The discovery of interference phenomena led to the development of the wave theory of light by Huygens (1690) and others. This theory accounted satisfactorily for the large-scale phenomena of geometrical optics, and provided such a beautiful explanation of so many additional effects (interference, dispersion, &c.) that by the end of the nineteenth century, the corpuscular theory of light was almost completely dis- carded, and the wave theory was universally accepted and was often thought to imply that light consisted of waves in a definite medium, the so-called ether. The wave phenomena theory, however, led to great difficulties when the quantum were discovered. In this class photoelectric effect and the Compton effect are seemed overwhelming that light of frequency of phenomenon- the examples---the evidence v gave up its en ergy definite quanta equal to hv, where h is Planck's constant. This ’In be- haviour seemed impossible’to reconcile with any咖ture in which the energy was associated with the whole of an expanding wave-surface, and clearly suggested the existence of light particles or photons with kinetic energy hv. There was thus an apparent conflict between the wave and particle properties of light, and the position was made more confused by the fact that all experimental attempts to detect the hypo- thetical“ether”resulted in complete failure. This conflict between the two theories of light was solved by a compromise which may be aummarized briefly as follows: (1)A light ray is to be regarded as a stream of moving particles or photons, light of‘ frequency，being composed of photons of kinetic energy kv. In this sense the old corpuscular theory is correct, and whenever we observe light we observe one or more whole photons. (2) When we wish to calculate the intensity of light, i.e.，the density of photons or the probability of finding a photon at a particular place, we use the equations of the wave theory with the assumption that the square of the amplitude of the waves is proportional to the light intensity. In an interference experiment, for example, the dark areas of low intensity are the regions where the wave equations indicate a small amplitude, and this is to be interpreted as a small probability of finding a photon at the place concerned. In this way the whole of the mathematical framework of the wave theory is retained, but we no logger imagine the existence of a vibrating medium. Light does not involve any actual physical vibration，
24 The General Background and although as a matter of convenience we still speak of light waves, wave-lengths, frequencies, &c.，what we really mean is that the in- tensities of light can be calculated by wave equations, with the inter- pretation that the square of the amplitude is proportional to the probability of finding a photon at the place concerned. (3) The interpretation of the wave theory in terms of probabilities applies not only to beams of light consisting of streams of photons, but also to the behaviour of an individual photon，and this implies that there is some degree of uncertainty in the behaviour of the individual photon. If, for example, we pass rays of light through a diffraction apparatus, we may obtain on a screen a series of bright and. dark rings， the former being at the places where the amplitude of the wave equation. is large and the latter where it is small. If nova- we pass one single photon through the same apparatus，we cannot predict where the photon will travel.All that we can say is that there are different probabilities of its travelling in different directions, these probabilities being greatest in the directions corresponding to the light rings observed when a stream of photons is passed through the apparatus. There is thus a certain indeterminancy in the behaviour of the individual photon， analogous to the behaviour of the electron according to Heisenberg's Principle. If, for example，we pass light through an aperture，then the smaller the aperture the greater is the diffraction effect, so that the more the aperture restricts the cross一section of the beam the greater is the uncertainty of the direction of the individual photon after passing through the apert ur e.Effects such as these can readily be described in terms of probabilities, but it is; of course, quite impossible to describe them in terms of either the older geometrical optics or in terms of the simple particles of the early corpuscular theory. The use of wave equationswi七h a probability interpretation does not enable us to “understand:「what is happening,”but it places us in a position in which we can describe and predict experimental results in terms of the appropri+te probabilities，and can include both the wave and particle aspects of light in one theory.、It will be noticed that this increased power is gained at the cost of an admission that we cannot always follow the individual photon in detail along its path. In the experiment ref+Jrred to above，for exaTi .ple, nothing enables us to predict which direction, the individual proton will take, after passing through the diffraetiol apparatus,,.We ma J calculate exactly the relative proba- bil反、:ic:oftho1)圣loton扭ovingind淤ren七.bilitncs opt t"o ),loton movlnig, 4.D- different curectlons一after passing through r}-J} ,}}+t}}，raft}`，O U, we there reach the Iimit of our powers, and the rest is inldetel}nind,te :}}c: unpredictable. 'rye ha vb ,}; .r: that electrons give rise to diffraction effects as though
5 2 The’Ideas of Wave肚“九a俪 Mechanics they were associated with waves, and it is therefore natural to enquire whether the wave-like and the particle aspects of an electron can be united in one single theory, as has been done for the case of a photon. Clearly, problem so far as the simple diffraction effects are can be approac卜ed in the same way, and we concerned, the may develop a “probability wave theory”of electrons on the as-sump Lion that the electron is a particle, and that the amplitude of a wave equation is a measure of the probability of finding an electron at the place concerned. The question then arises as to whether the wave一like- characteristics of electrons may affect their behaviour in other ways, and here an examina- tion of the relation、==一1is instru 刃吞u We know from experiment that Ve. an electron is accelerated or re- tard.---.d by an electric field, with a corresponding change in its momentum mu. From the relation h X二— 牡鑫u it follows that the wan。一length associated with the electron is inversely proportional to its momentum, and we can see, therefore, that as the electron accelerates, the wave- length will change. We have, thus, a condition of affairs in which as an electron moves through a field of varying potential，the associated wave-length is continually changing. Speaking generally, it pray be ,said that if the changes in potential are appreciable only over distances Which are large compared with the associated wave一length, the electron moves more or less as though it were a“particle”obeying the laws of“ordinary mechanics.”If, however, the changes in potential are appreciable over distances of the order of a wave一length, some com- pletely new phenomena are met with，and these cannot be explained in terms of the older mechanics.,In the :following pages we shall describe the general methods by which mechanics has been re一moulded so as七o take into account these wave-like eharacteristics of the electron, this branch of mechanics being known as“Wave Mechanics.”General consideratio xs of the magnitudes involved in the relation x== 立show 宁冷u that ,:be wave一len gths associated ;pith electrons in atoms are of the same order as the“Sides”of the atoms，or of the distances between atoms iri. metallic crystals. Under these conditions 4,-!he wave一I-}like character- istics of the electrons proci.uce many strange effects. a-J. some of these a ,re directly responsible for the properties of metai8。Many of these effects Plan be understood. only in terms of Lhe new mechanics, and if the metallurgist wishes to understand the. behaviour of electrons in metals, it is es`4ential for him to ua durstand the ideas which underlie wave mechanics. In the following pages we shall describe the main steps
2e The General Background or lines of argument by from the old to{ and the new means of which the mathematicians passed mechanics, but we shall present the results not concern ourselves with mathematical technique. We have adopted a rather formal method of approach, not merely because of its historical and logical interest, but because it serves to emphasize the correct way of regarding the wave-like characteristicsof electrons. The reader who does not wish to follow the argument may note that the method adopted is to re-write thein a form which is mathematically similar to certain e舞of mechanicsons of optical theory. In this way a problem of the motion of a beam of electrons in a varying field of force is made mathematically similar to a problem of the passage of a ray of light in a medium of varying refractive index. This involves the setting up of a differential wave equation. the inter- pretation of which is discussed in In the preceding chapters 下ve of Chapter VI. have referred to the circular orbits classical mechanics. These are 万功。3. merely one particular example of the more general problem of the motion of a particle. in a variable field of force, and we shall assume that the reader is familiar with the elementary equations of dynamics by means of which we can calculate the motion of a particle under different conditions of attraction, re- pulsiou, &c. These different equations may be expressed m more generalized forms, one Maupertuis (1698---1759)， of which and this is the Prime咖le of Least Action of may be understood in the following way. We may suppose that we wish to shoot a particle of given total energy B from a point .d to a point B (Fig. 3), subject to the condition that potential energy, W, and kinetic energy, K, may be interchanged, but that the total energy B -== W+K, remains constant. Then clearly the path which the particle will follow depends on the field of force in the space between A and B. In the absence of a field, the particle will travel认a straight line from A to B. In the presence of a画form field each as the gravitational field of the earth, we should have to “aim”the particle upwards from A，and the path followed would -be a parabola. The Principle of Least Action then states that if for all r8 conceivable Daths from左to万.we evaluate the integral ! zj%az, where‘ .A is the time, then this integral will have a minimum value*for the actual ’Strictly spewing we should refer to a stationary value rather than a min恤um value, but for a simplified description, the latter may be accepted.
行‘ O自 The Ideas mechanical .bath. That is to above integral for a series of of Wave Mechanics say that, if in Fig. 3 we evaluated the 由fferent paths I, 2, 3, 4…，then in the absence of a field of force，the m恤imhm value of the integral would be obtained for the straight-line path No. l，whilst the correct and仙.放叫 parabola would be predicted for the gravitational problem, similarly kith more complicated fields. For simple problem principle bas no advantage over the usual equations of motion, but complicated problems it is sometimes more convenient. 1r$ }2K& defines the quantity known as“action,'’and J通- The this hag the dim}xsions of MPT一 1. It will be noted also that the value of the rote gral depends only on the space co-ordinates of A and B, and not on the time. The minimum value of this integral for the actual mechanical path may be oafed S. The above remarks refer to the trajectory of a particle on passing from ore point to another, anti we may now consider the analogous problem of the passags of a beam of light from one point d to another B. In this case -we know that the path of the beam of light depends on the refractive ind。二，and hence on the velocity*of light in the space between the two paths. If the refracti :re index is uniform, light travels in a straight line from J. to Ajust as in the mechancal problem. the particle travelled in‘straight b.ne front A to B in the absence of a field of force. If th。refractive index is not uniforms. between A and B, the鞠ht'- travels in a path which is bent or curved, and which diepends on the t!xac} "};a .L isa;iou of the refractive index. The science of Geo- metrical Opti;Js de=lls with this Irind of problem, and the well-known 卜::，of refection axed refraction enable us to deduce the path of a beam 。吏吨ht when the, variation of refractive index and the positions of reflecting surfaces are known in the space surrounding A and B. Just as the Laws of Mechanics could be generalized in the form of the Principle of Least Action, so the Laws of Geometrical Optics may be generalized in a form kn。二as Fermat's Princsple of Least Time. This states that if in a problem where the variation of refractive index is known we consider a largo number. of conceivable optical paths of }fight travelling from A to B, then' the actual optical path light follows is characterized by。minimum t·value of the ，砌一. 尸1诬 for a beam which the integral: where v is the ve琳ity of light at the place ds, and ds is an element of *The velocities of light in different media are inversely proportional to the refractive indices of the different media. t See footnote to p. 26.
The Geneyal Background 0八︸ 9一 length of the pa:x.h.This implies that of all conceivable baths the light travels by the. path which enables it to reach刀0,6 uuickly as possible, and this principle e而bles the 'laws of reflection and refraction to be predicted. It will be seen, therefore, that there is a certain reseinblapce between the laws governing the:; trajectories of particles ire the rraechanical problem and the paths of beams of 'light in the optical uioblena，since in each casE th e actual path followed is characterized by the minimum value of an还tegral:Further, in a general way it can be seem. that the variation of the refractive index in the optical problem J_j'.ays the same part as the, field of force. or potential in the mechanical problem.This correspondence can be expressed in a precise mathematical form，so that the problem of the motion of a particle in a field of force bec:omes mathe- matically similar to the problem. of the path of a ray of light in a me击um of variable refractive index. These resemblances between the equations of mechanics and geometrical optics were noted many years ago by Hamilton，and were long thought to be nothing more than an interesting mathematical coincidence;it was only with_多}e work of tie Brogue and Schrcidinger that it was realized that a supremely fin- portant principle Jay concealed. So far we have described the correspondence between the mechanical Prof)p lem of the trajectory of a particle in a field of force and the optical ProblemofthePat1-problem of the patd Of a ray of light in a medium of varying refractive index, and we have discussed this in terms of the older geometrical optics. It is well known that the laws of geometrical antics can be expressed in terms of the wave theory of light, and all the equations for the paths of rays under conditions of reflection，refr}>>}t?1}}., ()(}(}r.，can be expressed in terms of the wave theory. For simple problems of geometrical optics, no advantage is gained by using the ware. theory, but when we deal with problems in which the refractive index varies appreciably over distances of the order of one wa孔-length，the wave theory preizicts a large number of new effects (dispersion, diffraction, &c.) which were quite inexplicable in terms of the older theory of rays. We see, therefore, t1iat the change from the older geome tri4ml optics to the wave theory did not affect what may be called large-scale pheno- mena (paths of rays, reflection, &c.)，but accoanted for a number of small-scale pheriomena which could not otherwise be understood. Since the charge from geometrical optics to wave theory does not a价ct the large-scale phenomena, the correspondence between the equations of rzebanics and those of geometrical optics to which we have referred above ca1l be expressed as a correspondence between the equations of mechanics and those of the wave theory of light. In this
The Ideas of Wave Mechanics 29 way it is possible to rewrite the equations of mechanics in forms which are mathematically similar to those of a wave theory. If this is done, the new equations naturally contain terms which play the same part that wave-lengths, frequencies, &e.，would play if the equations actually referred to a wave motion. The mathematical correspondence is so complete that for many purposes it isjustifiable and helpful to visualize the equations as though they were equations of a wave motion. In this way we may think of a moving particle as being associated with wave-lengths, frequencies, &e.，but we must remember that this is only a convenient way of visualizing the equations, and does not imply the e五stence of any physical waves. As stated above, it is possible to rewrite the equations of mechanics in forms which are mathematically similar to those of a wave theory. So long as we are dealing with large-scale problems, this procedure has no advantage, and it was at first looked upon as being little more than clever mathematical jugglery. The genius of de Brogue and Schrodinger lay in the fact that they were prepared to carry out the transformation of the equations of mechanics to those of a wave theory, and at the same time to hold fast to the correspondence between恤e effect of a field or force in the mechanical problem and of a varying refractive index in the optical problem, and to carry this correspondence to its logical conclusion. In the optical problem a variation of refrac- tive index which is appreciable over the distance of a wave-length produces effects which cannot be explained in terms of the older theory of rays. Hence it was suggested that in the mechanical problem a variation of force over very small distances might produce effects which could not be explained in terms of the older mechanics. But theso effects might be expressed还terms of the new equations, because these、 transformed equations were those of a wave theory, and a wave theory was able to deal with the effects observed when the refractive index varied appreciably over a distance of the order of a wave-length. In this way by rewriting the equations of mechanics in a form similar to that of a wave theory, and by retaining the correspondence between force and variation of refractive index, it was possible to build up a new mechanics, the so-called“wave mechanics," which predicted new effects when the force varied gre:.tly over very small distances and was thus directly applicable to the problem of the motion of an electron in the field of an atom or in the periodic field of a crystal lattice. It need scarcely be said that in its early stages this development of mechanics was regarded as little but mathematical speculation, but the whole position was transformed when .it was discovered that electrons showed diffraction effect s as though they were somehow associated with waves.
30 The.General Background This discovery showed that the- rewriting of the laws of mechanics in the form of equations of a wave theory。not just an example of mathematical ingenuity, but had revealed a real characteristic of the physical world. It is highly important for the student to realize that wave mechanics arose.before the discovery of the wave-like properties of electrons, and as the result of pure mathematical reasoning which showed how the equations of mechanics could be rewritten in the form of a wave theory. The student who wishes for further details will find these纽the Appendix on p. 32. For the general reader it is sufficient to note that wave mechanics came into being as the result of the line of argu- ment described above. When the equations of mechanics acre trans- formed into those of a wave theory, there 'are natura斯 relations between the terms of the wave equations and the dynamical properties such as momentum, energy, &c. In our comparison of the optical and mechanical problems we have dealt with the similarity between the laws governing the paths of particles and of rays of light, and have not considered the individual photons. Our first description should therefore refer to beams of electrons, rather than to single electrons or other particles. Historically itl was fit pointed out by de Brogue that if waves were associated veith particles, then the Principle of Relativity indicated that: -Energy_ Frequency一 Momentum Wave number 二Constant.I (9a) (9b) The constant was then intuitively identified with Planck's constant, h, and since the above relations were known to Main the case of light (energy hv, momentum hv/c) it was suggested that they might also apply to particles..‘The second of these relations was then confirmed experi- mentally by the electron-diffraction experiments of G. P. Thomson and of Davisson and Germer, “were diffracted by a crystal which showed that electrons of velocity 无_~ 叫Ml to ---.T Uonsidered ，肠肠 ength and momentum is the foundation of could therefore be concluded that )ossessed a wave-length i relation between wave- the whole subject. It the first of the above relations would also be satisfied, _ and the particles,加would be equal arguments of de Broglie showed that for to the relativity energy E. given by: .As explained 甲价了幼t. on p. 22, this relation req响res modifications if the veloci勺，is
了he Ideas 凡 of Wave mc. 一了- us1 -- 2c Mechanics 31 +t}1.…I (10) Which for ordinary problems with that of light is the same where the velocities are small compared as by=B.，二mc2+Jrnu2+可二I (11) 口口.，口口.、‘ The energy凡thus includes the energy mc2 associated with the mass of the particle. It may be thought curious that we should suddenly change from the ordinary total energy E二in+u2 -I一W, to the total energy凡，but it must be remembered that the potential energy, W, has always to be measured from some arbitrary zero, so that the addition of the constant term mc2 is merely equivalent to a shift in the zero from which the potential energy is measured, and consequently it does not affect the problems which we have been discussing. As will be appre- ciated from what follows the absolute value of，is never required,* and the physical properties dealt with involve either the differences between two frequencies or the differential of the frequency, for which the constant term mc2 is not involved.' we have, therefore, the two fundamental relations: E,s a 二AV 再 二二二一 们洛u 二I (1卿 I (12b) In the wave mechanics developed by Schrodinger, the essential assumption一made was that the waves were of a sine form, and it could then be shown that t瓜above two relations (12a) and(12b) were those necessary to make the mathematics self consistent. That is to say, it could be shown that if the hypothetical waves were of a sine form, in the equations of which the time‘entered as a term 2n-d, then the mathematical scheme became consistent if the frequency v were directly proportioyal to the energy.才In some books the reader will find the relation (1卿deduced in this way as the relation necessary to produce a consistent scheme, and in the same way the relation(12b), which has a direct experimental basis, fits into the general scheme. *Thus N. F. Mott(“An Outline of Wave Mechanics”)writes v = L'/h,， the problems with which he is concerned the change makes no difference alt the frequencies are enormously smaller than those梦，妙_，“Bn1h._ Thi serve如emphasize the purely symbolic and mathematical nature of the thetical waves. t The reader should be warned that in some books the same symbol i indiscriminately for R and A
and much confusion is caused in this way. t The general equation of a sine，.，.is of the form sin 2v(v‘一二/A+4
;63 2 The General Background We have, therefore, now carried our discussion to a point at which the mechanical problem has reached the second stage of the optical theory (p. 23), because the hypothetical waves have been associated with frequencies and wave-lengths which agree with the observed interference phenomena. From the hypot' :}tical above relations it will be seen that the velocity o} the waves is梦ven by: Em__ 口二二二V^二二二—二匕 分冷U MG2+蚤mug·十手于了 刃Z肠 I(13)t In most ordinary problems, the terns met is so much the, greatest that } is of the order 0/u, and since、can never be greater th。、、。，this implies that U must always be greater than c. This serves to emphasize*that the waves are merely mathematical functions.，and. are not of any material nature, because nothing material can tra-'El血utc r than light， The velocity U is called the phase-velocity of the waves, and gives the velocity with which the waves travel. V is thus greater than c, and consequently U cannot be the velocity of the ac Luai electron，since this cannot exceed c. We shall consider later what function gives the velocity of the electron itself, and for the present eve may say that the argument leads to the conclusion that moving electrons can be tfeated mathematically as though they were associated with waves of phase- velocity 62/u, where u is the velocity of the electron. · We have seen how the laws which express the motion of streams of electrons can be manipulated mathematically into a form which resembles that of the optical theory, and we have now to see how the electron density, i.e.，-probability of finding an electron at a particular place, is to be calculated, and, as in the case of the optical theory, the interpretation is that the amplitude of the hypothetical waves is a measure of the electron density at the place concerned. APPENDIX TO CHAPTERV. We have described_ above how the， Principle of Least Action enables the trajec- Cory of a particle to De predicted as tine pain which gives a mirdnitun value to tine integral r丑 JA for MI. For conveizience we may denote the minimum value of this integral the actual mechanical path by the symbol S. U y'e how in the optical problem the Principle of Least Time enables have also explained., ., P the pain or the ray 』犷万e卜ave repro呼uced above1"i . S专he只escri砂ion which， the red, .,呼er呷usually find7 黑Me Lext-DOOKS. n ll'tZle consideration w1u show that sirce the potential energy w has do be measured from some arbitrary zero, the frequency v cannot De uniquely defined Can fre the ，because W in equation I (11) may have any arbitrary conF.tant added to， it. ocher hand, tine cunerence Detween tw e了 reguencies and the山rrerential of the queney do not involve the constant term.
笑食 -gyp Ideas of Wave Mechanics of light to be rrcdicted as that which梦yes a minimum value to the integral The laws of veom,e tr; cal optics can be expressed ;n terms of the wave theory, and the Principle of Lea2t Time can be interpreted in terms ot this, tbeorv. Fc}. this purpose we ma Y again consider the passage of. a beam of light from A to B, :and ，B.J。 、.，_。.‘口GC a，。、，。。.抽.， we may denote Wl应 6 munimum value of口一，which characterlZes the actual optical 扩JA v-‘ path, by som e ordinates DA of Y at the A a ,crr^ bc L say Y.V and夕，so that if index it the surrouneing value of Y point五，say Yb. C surro-o nding space A，.S.1A that if狱 The value of Y is determined by the space co- 飞”，light吧呼from产and. travels to. B,呼叫”” a be cascuiarea u tine variation of the roxramive is known. it travelled Clearly there are many other points from A to the points concerned the woalr} }:Iso he YB, characterized by the:-fact, that if the ialrtc of I' wwudd be Ya. and. we can imagine,- . . r. > r t乎_」 终赞}ravet}ea from d tiv1 . u TFle surrounUM9 space surrounded by a surface any point on the surface, were homogeneous, this surface would be } sphere, wbilst if the refractive index were a variable the sh ，.，，，，、、1，.，。，.[ads[刀，，__1 oI Lne .C l3Gi'Z几 .'、3 0,v;,ula change accorazniv. olnce i—=i at, where f .L V J孟公j通 ape tune, this the time t 5orf二。。T.Vi.? represent the ends of all possible rays of瑰ht 一幸 —4,0!t to travel grow A sad travelling for the same time that the 沁B. In this way we can imagine the spa one ra又叮 ce rounli ti to coma a number of suriaMs Y,, Y2, Ys.，.corresponding to the position at tinges ‘=t1, t2, ts. &-c.:of the ends cf all possible rays of light leaving A at the time t -- #Q. In t h .}‘、-av: theory, the wave front is then regarded -as an expanding surface w:..ich:P }.}uds out from A parallel to the Y surfaces, and the rays of light are perpendicular tt: the wave 玩the uechanical problem sud兔ce. we have; denoted by 8 the minimum value of the. integral万2。: from A to B. It i$ bich characterizes the actual mechanical path of the partick ;。clear, therefore, that 8 in the mechanical problem eorrespondf with what w}. have called姗ical problem. Just as in thewe regarded A as surrounded by surfaces of constant Yy so inproblem we in.-'..v, revard A as surrounded by surfaces of constantof these being tl1 a.i} if the particle travehs from A to any -point opl the 8 On the meaDing the surface (.4 concerned, say召
, the value of the integral 一J通 2 Kdt is the same. We bave now to enquire whether a mathematical functign can be found which may be interpreted 吕习吕n expandint- surface spread.in that the wave s.1f face The九uction ch osea we con如p o Urseiv es by:. spreads out g out p parallel arallel to the S surfaces, in the same way to the Y surfaces in the problem. 铭 to called the action function, and is cases where the energy, E, remains co often S*.U is g1 d c'n AS*二J8一E(‘一to).……I (14) where t is the 1A me considered and to is In this way the which may be Principle of Least Action the time at which is combined with a the r以a .cle leaves A， tical function inte ae，expanding surface, and a further analogy with the eo汀 obtained. in which equation I(14) represents an exp surface nay be shown by。 pose glen cmisidcring a simple case where to二 0 and刃二 at S*二S一t. se we consider Lhe S surfaces surroundin gA，for which S二I加，101 at time f=0, tine surface S二 I00 corresponds with S* - 100. 102 A七 Sup- ...time t==I，S* -- 100一I==99, and it is the surface S== with the。 Mrfs,ce S*=100. Consequently. the surf" which now corresponds =100 has moved. out- ward, and similar in the time. 101. ，S* it nb7gical Y itli further increase.,I I iuterpretal.jon of One 纽ction frinction,a a is not possible to give acid historical.v it wa-,i as a. x i}cE} of mathematical iuzelerv. .t峥.J声.户- which was only show }a later to be of pra ma-the cticai si mat三 cal gn fin 沮caiice_ The action function mast therefore be looked on as a nction aspociated -mith a moving particle. D
}'ke General Background 4 0口 It oan also be shown that the momentum of the moving particle is perpendicular to the S* surface, just as the direction of the my of light was perpendicular to thewave mrface in the optical problem. .In the study of light, the wave theory and the older georaetrical optics leadto the same results for large-scale phenomena, and the " wave " aspect of lightonly becomes apparent when w e are concerned with distances of the same orderas that of the wave-length. In the above discussion. we have shown that thePrinciple of Least Action in the mechanical problem can be reconciled with theconcept of the spreading surface of the action function S*. The bold concept ofwave mechanics was then that this surface of the action function S* could be united with equations analogous to those of the wave theory of light, so that theS* surface could be regarded as associated with waves, although, just as in theoptical problem, these waves are not waves in any physical medium, but are tobe interpreted on a probability basis. -,As explained., on p. 31, the essential assump.tion made by Schrodinger was that the hypothetical. waves associated with theexpanding S* surface were of a sine form, and it could then be shown that themathematics became consistent if the fundamental relations of equations I (12a) and I(12b) were satisfied. The reader w进thus appreciate of waves with and was made the expanding S* surface was the result of mathematical before electrons had been shown to undergo diffraction
CHAPTER VI一`NAVE EQUATIONS AND TIjFIR INTERPRETATION. WF have now to proceed to the final stage of setting up a wave equation and interpreting it in terms of electrons. A beam of electrons of uniform velocity will correspond to a monochromatic ray of our hypothetical mechanical一waves, although the velocity u of the electrons /。2\ is totally different from the velocity(一C、of the waves;we shall con- \肠/ .cider later what function of the waves corresponds with the actual velocity of the electrons. The hypothetical“something‘ which vibrates”is denoted bar T, and in accordance with the general analogy with the theory of light, the amplitude of the hypothetical T vibrations is a measure of the electron density at the place concerned. A large value of the amplitude of the T vibrations corresponds with a high electron density and a large probability of finding an electron at the place concerned. In expressing this precisely, a difference exists between the optical theory and the wave mechanics. In the electro- magnetic theory of light, the state of the vibrating medium is described by two vectors,' B and H, and the intensity of illumination, or the density of photons, is proportional to (E2+护)，and this intensity fluctuates with the time，because .E and H are taken to describe waves in pbase with. one another. In the same way fi we may describe the hypothetical 'IF vibrations by two functions f and g, so that the electron density is proportional to the value of俨+尹)at the place concerned. The wave-mechanics theory is then developed on the assumption that at a given place(尹十92)，and hence the electron density, is independent of time." This implies that f and g must be one一quarter of a period apart. In the very simple case of a plane wave，for example: if f二a cos 2rc(vt一二/‘)……I (15) then g==a sin. 2n(W一x f a). So that户+92===a2 [c082 2n(vt一二/‘)+Si.n2 2n(vt一二/)I)1=a2a. *A vector is a quantity which has both a quantity magnitude and direction, and a scalar hands of a direction. involving clock magnitude only. Uthe radius ofa circle revolves like the length of the radius is a scalar a circ e revvl ves l}ke the quantity independent of 墩ent is taken from " An Outline of Wave Mechanics," by N. F. Mott(Cambridge University Press), with the permission of the author and publisher.$ The theory developed in this way- because when wave mechanics was inventedit was necessary to assume wave functions to describe the phenomena. Theessential requirement was a measure of the. intensity of electrons in a beam, andit was therefore convenient to adopt a function which made the intensity of abeam independent of the time at any one place. 35
36 Tote General Background In this way沪+尹)is independent of tine, and in this very simple case is also independent of the snac。一co一ordinate x, although in the more general case〔产+尹)varies frvrn place to place. The method universally employed is to replace the pair of real functions, f and g, by the single complex function: T=f+JlI..…，.I(ifi) where‘二A/只1. The conjugaUe complex o‘二which, is denoted T* is equal to f一ig, and the electron density is proportional to: 户+尹==tf十幼(.f一li9) = 'Pill'*。.工(17) The reader who is not familiar with the use of complex numbers may content himself with noting that the method used is merely a mathematical device for expressing the assumption that the hypothetical IF waves are the result of two vibratory processes which are one-quarter of a period apart, and so unite to give a resultant which is independent of time at any one place. The assumption often leads,she beginner into difficulties, because there is a natural tendency to think of a wave process as involving a fluctuation with time, but as can be seen from the above example of sine and cosine functions, it is quite possible for two vibratory processes to unite so as to produce a resultant w%ich is independent of the time. In the symbolism of this method, the electron density is proportional to the value of TT* at the place con- cerned, and the number of electrons in an element of volume△V is proportional to VNP*A V. It is helpful to remember that TT* is real, even though平be complex, and the electron density is always pro- portional to TV*.In some problems it is possible to express the electron density in other ways, and to understand this the details given in the Appendix (p. 39) must be appreciated. For a stationary state in which the energy is fixed and equal to hv, the general form of the function T may be written T (X, y, z, t)二0(二，y, z)CIMVJ.…I (18) where O(xyz) is sometimes called the space factor or the amplitude factor, and e2nir= is the time factor. This is merely another way of expressing the assumption that the electron density is independent of time, since e2MY9二cos 27rvt+ti sin 27rvt…I:19) which is of the form (f+ig) used in the previous description. 'the scalar value of e2rivl is thus unity, though this is not to be taken as
卿，‘ n公 Wave Ep ations and Their Interpretation implying that the vibratory process is independent of time;rather is it imagined to be the result of two vibrating processes, each of which fluctuates with time, but which compound ao that their resultant， although in the. general case varying from place to place，i} iv-dePendent of time at any one place. In equation I (18)，0 is thus the amplitude of T, and 7 and q/r have the same scaaar value，so that as long as we ate not co -1C;erned with the time, T and o are often interchangeable, since they have the same scalar value,tand obe v the same differenti- l equations (see bel。二)，and the electron density is thus proportional to 种. We have seen that the general problew. of the motion of a particle in a variable field of force is analogous to Vie motion of light gays in a medium of variable refractive index. In general the variation of the potential zrafirgy as a function of the spas; co一ordinates will be known, and the wave equation expressing the inagnifude of甲first proposed. by Schrodinger i of the form: VZ甲一卜 !,X1一14T) ' L=0…I (20) Here评is the potential energy。。d P the total energy, whilst the /1}2!!P〕ZW n2t护\ svmbol俨T stank;; for{奋妥+诀云+沃福)an"" is sometimes written 。J一--‘·--。---一\a X2。ay·az” /--------一-一’一 AT by Continental writers二This equation is known as the First Sc:hrobdinger Equation，or the tinze-free equation，for ordinary (non- relativity) mechanics, and it sponding to one value of E. red.=resent钾 ，As explain a family of surfaces, each corre- ed in the Appendix (p. amplitude中obeys an equation of exactly the same forra. 39) the In the present book we are not concerned with the mathematical development of these equations, or of the further equation described in the A.ppendix, but the reader will meet the equations so frequently that the meaning of the symbols should一e understood. In the general type of problem with which we are concerned, the variation of万is known一 thus, in -uhe problem of the hydrogen atom the potential·energy mechanics consist in 己2 is equal to一一一—and tae metnoas of wave finding suitable solutions eq-jation, and then interpreting these on to the appropriate the assumption that is proportional to the electron densit-,Y a t the place considered。 As we have already explained, so far as the paths trav,;ersed are, con- cerned，the effect of the wave mechanics is to reduce the problem of 一_It is this which leads to the unsatisfactory position in whit_ . 1 1共，symbols for._ 77 v 7 and 0 are often娜erc扛anged, or are not uastmgutshed,， so that只“reader， m蟹 find the same symbol with different meanLngs on adjacent pages or the game eoox.
38 The General Background the path of a beam of electrons in a field of varying potential to a problem of rays of light in a medium of varying refractive index. Examination of the equations then shows that with a particle of total energy, E, and potential energy, W, the effect of var户ng W is equivalent to the effect on a light ray produced by a varying refractive index.,“，given by: /，W\香 IL二(1一T,夕 \.乃/ In general, wave equations of the above type have large numbers of solutions, and the correct solution has to be chosen so that it represents something which is physically possible.Su命solutions may be called acceptable solutions, and in most simple problems they result in the condition that 0 shall be everywhere; finite and single-valued, and that both*and鬓shall be continuous. In most。一。where the solution represents a stationary state, the conditions to be satisfied result in the wave-length, a, being simply related to some length which the problem involves. Thus, if we consider an electron contained i n a one-dimen- sional box of length L, acceptable solutions are obtained only for which the wave-lengths are equal to 2L /n, where n is a whole number.* This in its turn implies that only certain energies can give rise to acceptable solutions, and these. energies are naturally interpreted as being those of the stationary states of the. older quantum theories, and the whole- number relations which the wave-lengths must obey are naturally regarded as connected with the older quantum numbers. In this way it may be said that quantum numbers and stationary energy states appear in wave mechanics, not as arbitrary postulates imposed on. one system of mechanics, but as the inevitable conditions which must be satisfied if the equations are to yield acceptable solutions. It is thus of particular fascination to see how wave mechanics, which by its very, nature deals with continuous quantities, results in the introduction of the whole numbers required to explain the quantum phenomena. * If a stretched wire is set vibrating, a steady vibration (i.e., a stationary state ofvibration) can be set up only for wave-lengths which are%definite fractions of thelength of the wire. The wave-lengths of the various possible steady vibrations arethen related by whole numbers. This is a physical vibration, but the mathematicswhich express the whole number relations are roughly analogous to those whichintroduce the whole numbers into the above problem of electrons. As previouslyemphasized., this does not mean that the electron is a physical wave.
9 nO Wave Equations and Their Interpretation Apr&NDTX TO CHAPTER VT. The relations implied by the use of complex quantities may be understood by the usual diagram in. which the real。 and imaginary quantities are plotted， at right angles, :.e., one-quarter or a revolution apart, lust as tine sine and cosine terms inT /1 .. , I. " .. . Y ." . T1. . .. . 手、妙洲are?钾-quarter of a。 revolution apart.，少拜瞥护咚毕，，袱MYg.气，POML弓.。马 Tine norizontai axis represent the real numbers ( j十us), wnust znose on the vertaoai represent the purely imaginary numbers (0+ig). bars '1 ''二，(,f+ig) and T* represent the co length OP equi .-补 The vectors OP and 二((f一ig), respectively. and is a real number, called the modules of T, .坦.下le .n伽尸b a:一OT and is written 171. The angle } is called the angle t of T, and to multiply twocomplex numbers the angles are added and the moduli are multiplied. It willbe seen, therefore, that on multiplying (f + ig) by (f - ig) the product will be apoint on the horizontal axis, i.e., a real number, of magnitude f2+ g$, so that $TT* = 1 q'' 1$, and in the problems with which we are concerned the electron densitymay be written as proportional to 1 T 1 s.We have explained above the general form of the function T as given, byequation I (18), and have seen how this contains both an amplitude factor O(x, y, x) and a time factor eta4vt. If the time factor then this equation is chosen so that T is of the form given by real, and equation I(18), represents a progressive wave system if功has a suitable cam wave if价is It is common practice to omit the time factor in equation I (18), and tions, such travelling 。奋八od that the reader will often encounter equa- as功二ea", described as waves, and in such cases representing .it is urider- the time factor is to be added. As explained on p. 37, the electron density is proportional_ to种*，but in many problems con- cerning， stationary states, 0 is real， and_师 * isI --.+ t ".t to 7t t t t }} " "t t identical with yr. in such problems It is posshbie to draw a graph sh distan ce in a 功varies with and from the this gra户一the electron density can be read off by squaring ' the values of e}. It is only in someproblems concerning stationary states that thisis possible, and the reader should note that 00* and e are not always interchangeable, and thatthe electron density is always proportional to The use of imaginary quantities in the abovet .I ." t "1 t "-t way is merelyt，a， mathematical device which.1 .0 P 011 redu,.卿，some of the equations照a form witint t wnicn tine mathematiLum场laminar, and when f Fio. 4. in sure result of two vibratory processes, which are one- s the above treatment merely assumes that the hypotheti combine to give a resultant independent -of time at any on above, the first Schriidinger equation, I (20), represents a t， This angle is often called the amplitude of the complex number, bu avoided the use of this expression. in order to prevent possible CODIUSIU}”- l八 一扮1万 V. the amplitude of the T waves. t This is sometimes expressed by saying that the scalar value of T is叮耳了一 This is correct, but should not lead one to write'F2 instead of JlF1=. 怪Confusion has arisen in many of the books owing to the fact that the descrip= tion says that the electron density is proportional to种介，whilst the examples dealt with are stationary states of the above. type, with the result that the beginner is puzzled by the change from师*to护·
4O The Gen eral Background ‘vrfa“兮‘粤咚尹r呷卿山移t.oon”v叶usurfaces each corresponding; to one a given by t(16). the tune factor may De 。n6vivu e of E. U甲is assumed to be of the form V20+ S沪.m ,4$ cancelled out, and the amplitude equation (E一W冲=0..…1(21) is obtained, so t恤t as stated before, both `k' and功obey the same differential equation. If the time factor has the same form as in be eliminated from equation I (21)to yield the Second I(18), the energy E may Schrodinger Equation: V21:一8擎二十 几. 4vni云甲 1(22) This is the more fundamental equation, and can iR an eaulicit furtetion of the time. Those who 刁心 be used for problems where评 wish for further information on or use of the equations should consult the following books: .Elementary. Dushrnan, S.“Elements of Quantum Mechanics." New York:1928 (John Wiley&Sons,玩叼.____________ Flint, H. T.“Wave Mechanics." 6th Edn. London:1946 (Methuen & Uo., Ltd小 “Elementary Quantum Mechanics." Cambridge:1934 (University “Wave Mechanics." Dublin (Dublin Institute for .Advanced “An Outline of Wave Mechanics." Cambridge:1980 (University Frenkel, J. Press). Frenkel, 3. Press). -More Advanced. “Wave Mechanics:Elementary Theory." Oxford (Clarendon “Wave Mechanics:Advanced General Theory." Oxford (Clarendon Pauling, L.,; New York: Rojansky, ti'. and Wilson, E. B.,打. “Introduction to Quantum Mechanics." 1935 (McGraw-Hill Book Co., Inc.). “Introductory Quantum Mechanics." New York (Prentice and Hall).
CHAPTER VII.---THR REPRESENTATION OF THE INDIVIDUAL ELECTRON. IN the preceding desc帅tion we have dealt with the rays or beams consisting of a continuous stream of electrons, and the wave function was interpreted so that TT*A V was a measure of the probability of finding an electron还the particular element△V concerned. We have next to consider what happens when we try to identi勿the individual electrons in the stream, and here the interpretation offered is that the total Y'' is regarded as the sum of the effects due to the individual electrons, so that TT*A